Mathematical Research Letters

Volume 16 (2009)

Number 6

A note on compact Kähler-Ricci flow with positive bisectional curvature

Pages: 935 – 939

DOI: http://dx.doi.org/10.4310/MRL.2009.v16.n6.a2

Authors

Huai-Dong Cao (Lehigh University)

Meng Zhu (Lehigh University)

Abstract

We show that for any solution $g_{i\bar j}(t)$ to the Kähler-Ricci flow with positive bisectional curvature $R_{i\bar i j\bar j}(t) >0$ on a compact Kähler manifold $M^n$, the bisectional curvature has a uniform positive lower bound $R_{i\bar i j\bar j}(t) >C >0$. As a consequence, $g_{i\bar j}(t)$ converges exponentially fast in $C^{\infty}$ to a Kähler-Einstein metric with positive bisectional curvature as $t\to \infty$, provided we assume that the Futaki-invariant of $M^n$ is zero. This improves a result of D. Phong, J. Song, J. Sturm and B. Weinkove \cite{PSSW} in which they assumed the stronger condition that the Mabuchi K-energy is bounded from below.